Similar to the simulation setting of [1]}, the random scores, \(\xi _{nr}\) 's, are independently drawn from one of the three distributions: Nearly Uniform (NU) distribution, for which \(\xi _{nr}\) 's are drawn independently from standard normal distribution \(N(0,1)\) ; Moderately Nonuniform (MN) distribution, where the scores \(\xi _{nr}\) 's are independently generated by t-distribution with three degree of freedom and scaled to unit variance; and Very Nonuniform (VN) distribution, where \(\xi _{nr}\)  follows t-distribution with one degree of freedom and scaled to unit variance. Notice among these three distributions, the score distribution for NU has the lightest tail, while has the heaviest tail for VN. As a result, the functions in a synthetic dataset from NU will behave more homogeneously. On the other hand, the synthetic datasets from MN or VN are likely to contain some observations with extreme magnitude. [1]} note observations in the VN setting could have drastically different leverages, adversely affecting the naive uniform sampling.
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